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Few materials undergo cooperative Jahn-Teller (JT) transitions at low temperatures, but zircon-type oxides are one class that includes DyVO$_4$, which transforms from a tetragonal to an orthorhombic structure at around 13.6 K, with a narrow transition temperature range within 0.5 K. Since many rare-earth ions can be accommodated in the structure, there should be ample routes to vary the transition temperature and structural effects of the transition. We have synthesized pure DyVO$_4$ and solid solutions Dy$_{1-x}$Tm$_x$VO$_4$ ($x$ = 0.05, 0.1, 0.15, 0.2, and 0.5) and Dy$_{1-y}$Pr$_y$VO$_4$ ($y$ = 0.03, 0.05, 0.1, 0.2, and 0.5), all by solution precipitation. X-ray diffraction shows a systematic peak shift and a linear change of lattice parameters with increasing substitution. We demonstrate through heat capacity measurements that both Tm$^{3+}$ and Pr$^{3+}$ substitutions cause a depression of the cooperative JT transition temperature. When the substitution level increases, the JT transition is eventually suppressed. We examine a mean-field approximation model that can explain the both the JT transition temperature shift and its eventual disappearance. The cooperative JT effect in these zircon-type oxides is known to exhibit a large magnetoelectric response, which should follow from the mean-field behavior. The structural transformation can be easily detected via diffraction and can be used as a temperature calibration in low temperature experiments.